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A216419
Odd powers that are not prime powers.
1
225, 441, 1089, 1225, 1521, 2025, 2601, 3025, 3249, 3375, 3969, 4225, 4761, 5625, 5929, 7225, 7569, 8281, 8649, 9025, 9261, 9801, 11025, 12321, 13225, 13689, 14161, 15129, 16641, 17689, 18225, 19881, 20449, 21025, 21609, 23409, 24025, 25281, 25921, 27225
OFFSET
1,1
COMMENTS
Numbers in A075109 but not in A000961.
Also odd perfect powers having no primitive root (intersection of A075109 and A175594).
LINKS
Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Root.
FORMULA
Sum_{n>=1} 1/a(n) = 1/2 + Sum_{k>=2} mu(k)*(1-zeta(k)*(2^k-1)/2^k) - Sum_{p prime} 1/(p*(p-1)) = 0.0158808884... - Amiram Eldar, Dec 21 2020
EXAMPLE
81 = 9^2 as well as 81 = 3^4, therefore 81 is not a term.
225 can be expressed so in one way as (3*5)^2, therefore 225 is a term.
MATHEMATICA
nn = 27500; lst = Union[Flatten[Table[n^i, {i, Prime[Range[PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]]; Select[lst, OddQ[#] && ! IntegerQ@PrimitiveRoot[#] &] (* Most of the code is from T. D. Noe *)
PROG
(Magma) [n : n in [3..27225 by 2] | IsPower(n) and EulerPhi(n) ne CarmichaelLambda(n)]; // Arkadiusz Wesolowski, Nov 09 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved